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Perhaps Sylvester overstated the case, but there was a lot in
what lie said. If he did not exactly rise from the dead he at least
got a new pair of lungs: from the hour of his meeting with
Cayley he breathed and lived mathematics to the end of his
days. The two friends used to tramp round the Courts of Lin-
coln's Inn discussing the theory of invariants which both of
them were creating and later, when Sylvester moved away,
they continued their mathematical rambles, meeting about
halfway between their respective lodgings. Both were bachelors
at the time.
The theory of algebraic invariants from which the various
extensions of the concept of invariance have grown naturally
originated in an extremely simple observation. As will be noted
in the chapter on Boole, the earliest instance of the idea appears
in Lagrange, from whom it passed into the arithmetical works
of Gauss. But neither of these men noticed that the simple but
remarkable algebraical phenomenon before them was the germ
of a vast theory. Nor does Boole seem to have fully realized
what he had found when he carried on and greatly extended the
work of Lagrange. Except for one slight tiff, Sylvester was
always just and generous to Boole in the matter of priority, and
Cayley, of course, was always fair.
The simple observation mentioned above can be understood
by anyone who has ever seen a quadratic equation solved, and
is merely this. A necessary and sufficient condition that the
equation aa3 -f 2bx -|- c = 0 shall have two equal roots is that
bz  ac shall be zero. Let us replace the variable x by its value
in terms of y obtained by the transformation y  (px + <?)/
(rx -f s). Thus x is to be replaced by the result of solving this for
a?, namely x  (q  $y)!(ry  p). This transforms the given
equation into another in y; say the new equation is Ay2 + 2By
-f C = 0. Carrying out the algebra we find that the new coeffi-
cients A> B, C are expressed in terms of the old a, b, c as follows,
A = as-  Zbsr + crz,
B =  aqs + b(qr + sp) - cpr,
C = aq*  2bpq -f cp2.
From these it is easy to show (by brute-force reductions, if